Invariants
Base field: | $\F_{2}$ |
Dimension: | $6$ |
L-polynomial: | $( 1 - 2 x + 2 x^{2} )^{2}( 1 - 5 x + 12 x^{2} - 20 x^{3} + 29 x^{4} - 40 x^{5} + 48 x^{6} - 40 x^{7} + 16 x^{8} )$ |
$1 - 9 x + 40 x^{2} - 116 x^{3} + 249 x^{4} - 432 x^{5} + 648 x^{6} - 864 x^{7} + 996 x^{8} - 928 x^{9} + 640 x^{10} - 288 x^{11} + 64 x^{12}$ | |
Frobenius angles: | $\pm0.0635622003031$, $\pm0.165221137389$, $\pm0.250000000000$, $\pm0.250000000000$, $\pm0.365221137389$, $\pm0.663562200303$ |
Angle rank: | $2$ (numerical) |
Jacobians: | $0$ |
This isogeny class is not simple, primitive, not ordinary, and not supersingular. It is principally polarizable.
Newton polygon
$p$-rank: | $4$ |
Slopes: | $[0, 0, 0, 0, 1/2, 1/2, 1/2, 1/2, 1, 1, 1, 1]$ |
Point counts
Point counts of the abelian variety
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ |
---|---|---|---|---|---|
$A(\F_{q^r})$ | $1$ | $5275$ | $314509$ | $55519375$ | $1759622051$ |
Point counts of the (virtual) curve
$r$ | $1$ | $2$ | $3$ | $4$ | $5$ | $6$ | $7$ | $8$ | $9$ | $10$ |
---|---|---|---|---|---|---|---|---|---|---|
$C(\F_{q^r})$ | $-6$ | $4$ | $12$ | $36$ | $49$ | $46$ | $127$ | $260$ | $444$ | $1019$ |
Jacobians and polarizations
This isogeny class is principally polarizable, but does not contain a Jacobian.
Decomposition and endomorphism algebra
All geometric endomorphisms are defined over $\F_{2^{20}}$.
Endomorphism algebra over $\F_{2}$The isogeny class factors as 1.2.ac 2 $\times$ 4.2.af_m_au_bd and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:
|
The base change of $A$ to $\F_{2^{20}}$ is 1.1048576.dau 2 $\times$ 2.1048576.dth_ibxft 2 . The endomorphism algebra for each factor is:
|
- Endomorphism algebra over $\F_{2^{2}}$
The base change of $A$ to $\F_{2^{2}}$ is 1.4.a 2 $\times$ 4.4.ab_c_ai_z. The endomorphism algebra for each factor is: - 1.4.a 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
- 4.4.ab_c_ai_z : 8.0.13140625.1.
- Endomorphism algebra over $\F_{2^{4}}$
The base change of $A$ to $\F_{2^{4}}$ is 1.16.i 2 $\times$ 4.16.d_bm_bk_yb. The endomorphism algebra for each factor is: - 1.16.i 2 : $\mathrm{M}_{2}(B)$, where $B$ is the quaternion algebra over \(\Q\) ramified at $2$ and $\infty$.
- 4.16.d_bm_bk_yb : 8.0.13140625.1.
- Endomorphism algebra over $\F_{2^{5}}$
The base change of $A$ to $\F_{2^{5}}$ is 1.32.i 2 $\times$ 4.32.a_ad_a_bwv. The endomorphism algebra for each factor is: - 1.32.i 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
- 4.32.a_ad_a_bwv : 8.0.13140625.1.
- Endomorphism algebra over $\F_{2^{10}}$
The base change of $A$ to $\F_{2^{10}}$ is 1.1024.a 2 $\times$ 2.1024.ad_bwv 2 . The endomorphism algebra for each factor is: - 1.1024.a 2 : $\mathrm{M}_{2}($\(\Q(\sqrt{-1}) \)$)$
- 2.1024.ad_bwv 2 : $\mathrm{M}_{2}($4.0.3625.1$)$
Base change
This is a primitive isogeny class.